OFFSET
0,5
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
FORMULA
Number triangle: T(n, k) = Sum_{j=0..n} (-1)^(n-j)*C(n, j)*(-2)^(j-k)*C(k, j-k).
T(n, k) = Sum_{j=0..n} Sum_{i=0..k} C(k, i)*C(n+k-i-j-1, n-k-i-j)*(-1)^(n-k).
T(n,k) = T(n-1,k-1) - 2*T(n-1,k) - T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0)=-1, T(1,1)=1, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Jan 12 2014
EXAMPLE
Rows begin
1;
-1, 1;
1, -4, 1;
-1, 9, -7, 1;
1, -16, 26, -10, 1;
-1, 25, -70, 52, -13, 1;
MATHEMATICA
T[n_, k_] := Sum[(-1)^(n - j)*Binomial[n, j]*(-2)^(j - k)*Binomial[k, j - k], {j, 0, n}]; Table[T[n, k], {n, 0, 20}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 29 2017 *)
PROG
(PARI) for(n=0, 20, for(k=0, n, print1(sum(j=0, n, (-1)^(n-j)*binomial(n, j)*(-2)^(j-k)*binomial(k, j-k)), ", "))) \\ G. C. Greubel, Aug 29 2017
CROSSREFS
KEYWORD
AUTHOR
Paul Barry, Jul 24 2005
STATUS
approved